ath is all around, and an important part of your
life. You use math when you are playing
games, cooking food, spending money, telling time,
reading music, and doing any other activity that uses
numbers. Even finding a television station uses math!
Division Is Everywhere
You use division in everyday life. When you split a
bag of candy or a box of cookies with a friend, you
are dividing. When you decide how many rows of
seats your class will need during an assembly, you
are dividing. Any time you have a group of things
and need to separate them into smaller equal
groups, you are dividing.
Using This Book
This book can be used to learn or review division
at your own speed. It can be used on your own
or with a friend, tutor, or parent. Get ready to
discover math . . . made easy!

5

What Is
D

ivision is a way to break something into
equal parts.

You have 20 king cobras. A basket holds
5 cobras. How many baskets do you need
to hold all of your king cobras?
There are two ways to find the answer.
1 You can find the number of baskets by subtracting
5 cobras at a time from the total.

20 5 15

1 basket

15 5 10

2 baskets

10 5 5

3 baskets

550

4 baskets

You need 4 baskets to hold all 20 king cobras.

6

Division?
2 You can divide. Division is just another way to
subtract over and over again.

uppose you know a total number of items.
You also know the number of groups you
want to equally put them into. You can divide to
find how many items will be in each group.
Three friends bought a pizza. The pizza had
six slices. If three friends shared the pizza
equally, how many slices did each friend eat?

Six slices of pizza, divided equally among three
friends, gave each friend two slices.
Six divided by three is two.

63=2

8

Divide?
Suppose you know the total number of items, and
how many are in each group. You can divide to find
the number of groups.
You have ten bees. You want to divide them
into groups of two bees. How many groups
can you make?

You can make 5 groups of 2 bees.
Ten divided by two is five.

10 2 = 5

9

Division
S

uppose you have 18 pencils. You want to
divide them into groups of 6. How many
groups will you have?
The number 18 is the dividend.
It is the number being divided.

The number 6 is the divisor.
It is the number you divide by.
The answer to a division
problem is called the quotient.
If you make groups of 6 pencils,
you will have 3 groups.

Multiplication
Three rows of four chairs
make a total of twelve chairs.

3 4 12
Twelve chairs divided into
rows of four chairs each
will fill three rows.

12 4 3

13

Division
Y

ou can learn division facts by using the
multiplication facts. Letâ&#x20AC;&#x2122;s review the terms for
multiplication.

factor factor product
3 7
21
Letâ&#x20AC;&#x2122;s look at the same fact, written using division.

product factor factor
(dividend) (divisor) (quotient)
21
7
3
When you know the multiplication fact 3 7 21,
then you also know the division fact 21 7 3.
Remember: In
you
multiplication,
e
can change th
ctors,
order of the fa
er is
and the answ
the same.

60
The problem is asking, How many groups of zero
items will make six items? You could have as
many groups of zero as you want and still never
have a total of six.
On the other hand, zero can be divided by any
number. The answer will always be zero. If you have
nothing (zero) and divide it into any number of equal
groups, there will be nothing (zero) in each group.

02=0
0 27 = 0
0 312 = 0

16

and One
Any number divided by one stays the same number.
If you have eight baseballs
and divide them into one
equal group, the group will
have eight baseballs in it.

818
When any number is divided by itself, the answer
is one.
If you have nine beads and
divide them into groups of
nine, how many groups will you
have? You will have one group.

991

17

Long
S

ometimes you need to divide numbers that are
not basic facts. You can use long divison.

42 3

is written as

342
.

Divide one place value at a time, beginning on the left.
divisor

342
dividend

Divide 3 into the first digit of the
dividend (42).

How many groups of 3 can you make from 4? Find
the 3s multiplication fact with an answer closest to
4, but not greater than 4.
313

326

There is one group of 3 in 4. Two groups of 3 is
greater than 4.

1
342
3

18

Write a 1 in the answer above the 4.
Multiply the number you just wrote (1) by
the divisor (3).
1 3 3. Write the 3 below the 4.

Division
1
342
3
1

Subtract. 4 3 1. In long division,
the difference (1) must be less than the
divisor (3). 1 is less than 3.

1
342
3
12

Bring down the next number (2).
Now you have the number 12.
Divide 3 into 12. (Think: How many
groups of 3 can I make from 12?)

14
342
3
12
12
0

The number 12 will make exactly
4 groups of 3 (4 3 12).
Write a 4 above the 2.
Multiply. 4 3 12.
Write the product (12) below the 12.
Subtract. 12 12 0.
There are no more numbers to bring down.
You are finished!

42 3 14

19

Dividing Greater
Y
divisor

ou can divide any number by a one-digit
number using long division.

6294
dividend

6294

4
6294
24
5

20

294 6

is written as

6294
.

In the dividend, the first number on
the left is 2. The number 2 is smaller
than 6, so you cannot make any groups
of 6 from 2.
The next number is 9. Since you did not
use the 2, the number you are looking
at is 29. The number 29 will make 4
groups of 6 (4 6 24), but not 5
(5 6 30).

owers of ten are any numbers that are a one
followed by zeros. For example, 10, 100, and
1,000 are all powers of ten.
To divide by a power of ten, move the decimal
point one place to the left for each zero at the end
of the number.

50 10 =
50.
In whole numbers, the decimal
point is at the end. Write the
decimal point
dividend with a decimal point at the end.
You are dividing by 10, which has
one zero. So, move the decimal
point one place to the left.

50 10 = 5

28

50.
5.0

of Ten
35,000 100 =
Write a decimal point after the dividend.

35,000.
decimal point

You are dividing by one hundred,
which has two zeros. So, move
the decimal point two places
to the left.

35,000.
350.00

35,000 100 = 350

29

Multiples
W

hen both the dividend and the divisor end in
zeros, you can take the same number of zeros
off each and then divide.

250 50
Both numbers end in one zero.
Take one zero off each number.

250 50
250 50
25 5

Now divide. The result is the
same as the answer to the
orginal problem.

Estimation
You can estimate the answer to a division problem
with larger numbers by rounding each number to its
greatest place value.

207 19
If you round 207 to the hundreds place, you get 200.
The greatest place value in 19 is the tens place, so
19 rounds to 20.

200 20

The numbers 200 and 20 each end in zero. You can
reduce the numbers to make the problem easier.
The answer for 200
answer for 20 2.

20

is the same as the

20 2 = 10
207 19 is about 10.

33

Properties
A

re the same properties true for division that are
true for multiplication? Letâ&#x20AC;&#x2122;s check.

When you multiply, you can change the order of the
numbers and the answer will not change. Is this true
for division?

93

39

How many groups of 3 can you make from 9? 3

933
How many groups of 9 can you make from 3? None!
The number 9 is bigger than 3.
You cannot change the order of the numbers in a
division problem.

34

and Division
When you multiply more than two numbers, you can
group the numbers in any way to multiply them. Is
this true for division?

(60 6) 2

60 (6 2)

Divide inside (60 6) 2
parentheses first. (10) 2
Divide again.

10 2 5

(60 6) 2 5

60 (6 2)
60 (3)
60 3 20

60 (6 2) 20

You cannot group the
numbers any way
you choose in a
division problem.

35

Checking
B

ecause multiplication and division are opposite
operations, you can use multiplication to check
the answer to a division problem.
On page 19, we found that

42 3 14
If 42 3 14 is correct, then 14 3 42.
1

14
3
2
Multiply ones.
3 4 12.
Regroup.

1

14
3
42
Multiply tens. 3 1 3.
Add. 3 1 4.

14 3 42, so 42 3 14.

36

Your Answer
When the answer to a division problem
r emainderâ&#x20AC;&#x201D;Th
e
has a remainder, first use multiplication, number
left
then add the remainder.
over in a
division proble
On page 23, we found that
m.

30 9 = 3 R3
divisor

Multiply the divisor and the
part of the answer that is not
the remainder.

9 3 27
remainder

Add the remainder to the number
you just found.

27 3 30

The result (30) is the same as the number you divided,
so the answer is correct.

30 9 = 3 R3

37

Division
O

wen is half as old
as his sister Lauren.
Lauren is 14.
How old is Owen?
The word half tells
you that you should
divide by 2 to find Owenâ&#x20AC;&#x2122;s age.
14 years 2 7 years
Owen is 7 years old.

Words that help you know how to solve problems
are called key words. Some key words for division
problems are listed in the table below.

Division Key Words
average
cut
divided
divisor
each

38

equal parts
evenly
every
half
out of

per
quotient
separate
shared
split

Key Words
Some division problems will ask you to change a value
for more than one item to a value for one item.
If you know the:

and you need to find the:

Price of more than one

Price of one

Size of more than one

Size of one

Length of more than one

Length of one

A package of three computer games sells for
$51. What is the cost per game?
Since you know the cost of three (more than one)
games, you can find the cost of one game by dividing.
First write the problem in words.
cost of three games divided by three is the cost per game

$51 3 $17
The cost is $17 per game.

39

Word
W

ord problems can be solved by following
four steps.

Your club bought 168 flowers to give to
12 teachers. Each teacher will get the same
number of flowers. How many flowers will
each teacher get?

40

Problems
1 Understand the problem.
Find what you know.
Find what you want
to know.

There are 168 flowers and
12 teachers.
The number of flowers each
teacher will get.

2 Plan how to solve the problem.
You are looking for the equal number of flowers
each teacher will get. You know the total number of
flowers and the number of teachers. This is a
division problem.
3 Solve the problem.
Divide the number of flowers by the number of
teachers.
168 flowers 12 teachers 14 flowers per teacher.
4 Check your work.
Check division by multiplying.
14 flowers per teacher 12 teachers 168 flowers.

41

Understanding
W

hen a quotient has a remainder, you must
decide what the remainder means.

Letâ&#x20AC;&#x2122;s look at some problems that use the same facts,
but ask different questions.
A coin collection book displays four coins on
each page. You have collected 51 coins. How
many pages will you use if you put all of the
coins in the book?
You are being asked to divide your 51 coins into
groups of 4.

51 4 12 R3
The question asks how many pages will you use.
Twelve pages will be full; another page will be
needed for the remaining 3 coins.
Since 12 1 13, you will use 13 pages.

42

the Remainder
A coin collection book displays four coins on
each page. You have collected 51 coins. How
many pages can you fill completely?
In this problem, only the full pages are counted. You
will have 12 full pages.
A coin collection book displays four coins on
each page. You have collected 51 coins. How
many coins will be on the last page?
In this problem, the question asks how many coins
will be on the last page, which is the remainder.
There will be 3 coins on the last page.

43

Word Problems
L

etâ&#x20AC;&#x2122;s use the steps from pages 40 and 41 to solve
a word problem that uses a remainder.

There are 2,153 chocolate candies
in a bag. You get to eat
what is left after the
candies are divided
equally into 20 bags.
How many candies are
you allowed to eat?

1 Understand the problem.
Find what you know.

There are 2,153 candies
to go equally into 20 bags.

Find what you want
to know.

The number of candies
left over.

44

and Remainders
2 Plan how to solve the problem.
A large group of candies is being divided equally into
smaller groups. This is a division problem.
3 Solve the problem.
Divide the number of candies by the number of bags.

2,153 20 107 R13
The question asks how many candies you get to eat.
Since you get what is left over, or what remains, the
remainder is the answer. You get to eat 13 candies.
4 Check your work.
When division problems have a remainder, first
multiply (107 20 2,140), then add the
remainder (2,140 13 2,153). You are back
to the original number you divided, so your answer
is correct.